On iterations of Misiurewicz's rational maps on the Riemann sphere

A NNALES DE L ’I. H. P., SECTION A

P. G RZEGORCZYK

F. P ^RZYTYCKI W. S ^ZLENK

On iterations of Misiurewicz’s rational maps on the Riemann sphere

Annales de l’I. H. P., section A, tome 53, n

^o

4 (1990), p. 431-444

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

431

On iterations of Misiurewicz’s rational _maps

^on

the Riemann sphere

P. GRZEGORCZYK, F. PRZYTYCKI and W. SZLENK Polish Academy ^of Sciences, Institute of Mathematics, ul.

Sniadeckich 8, ^00-950 Warszawa, ^Poland Warsaw University, Department ^of Mathematics,

PKiN _{9 p.,} 00-901 Warszawa, ^Poland

Vol. 53, ^n° 4, 1990, Physique théorique

ABSTRACT. - This paper ^concerns

ergodic properties

of rational maps of the Riemann

sphere,

^of

subexpanding

behaviour. In

particular

^we prove the existence of an

absolutely

continuous invariant _measure,

study

^its

density

^and prove the metric exactness..

This is not a

strictly

^research paper. Its aim is to present useful facts

belonging

^{to the} folklore but

partially

^not

explicitly published

up ^{to now.}

INTRODUCTION

Let f

^{be a rational} map of a Riemann

sphere f,

let J denote its Julia set, y denote the

normalised,

standard Riemann measure on

ê

Derivatives in the paper ^are considered

usually

^with respect ^to the Riemann metric.

Annales de l’Institut Henri Poincaré - Physique théorique - ^{0246-0211 1}

Vol. 53/90/04/43 1 /14/$3.40/ © Gauthier-Villars

In this text we

verify

^{under suitable}

assumptions

^two

properties:

The property

(ii)

is called metric exactness. Satisfaction of the both

properties

^without any additional

assumptions

^{is a famous} open

problem;

we do not make any progress in

solving

^it.

Consider also the property

In section 2 we prove

(i)

^and

(ii’)

^{under the}

assumption

that all critical

points

^{in J are}

eventually periodic.

^{In section 3 we} consider the case where Crit J

== 0

^{and moreover}

f

^is

expanding

^{on J n the set of}

neutral rational

periodic points } ^{(i. e.}

there exists k &#x3E; 0 such that for _every

z ~ ( f k)’ (z) ~ &#x3E; 1 ).

^{We call}

^{such f}

Misiurewicz’s _map

by

^the

analogy

^with

maps considered in

[M].

^In

paragraph

^{4 in the case we} prove the existence of an invariant measure

equivalent

to ~

[this

^{allows to} pass from

(ii’)

^to

(ii)]

and prove that outside the set it has

real-analytic

subharmonic

density.

Everything

^{in this} paper is the so called folklore for

specialists,

^but

partially

^not

explicitly published.

^{So we decided that} this material is

worthy

^of

being

written down. Lots of facts from this _paper

rely

^on

Koebe’s distortion

theorem,

^see

[H],

^{Th. 17.4.6.} One version is that for every ^t 1 and C &#x3E; 0 there exists

C (t) &#x3E; 0

^{such that for} every

holomorphic

univalent

function f : ^{[]) --t ê}

^{on the unit}

^disc,

^{such that diam}

(~B~f ([])))

&#x3E; C

Contrary

^{to a}

recently popular

^{custom no} involvement of a

hyperbolic

metric is necessary. Our ^text is

elementary.

^{Let us call the} attention of the reader to

significant Mary

Rees’ paper

[R].

^Our considerations consti- tute

only

^a

starting point

^{to her}

infinitely

^{more delicate}

study.

^A part ^of

our paper

overlaps

^{also with M.}

Lyubich

paper

[L].

This text grew from discussions at the Cornell conference on iterations in

August 1987,

^{a talk of the second author at} the Warsaw

dynamics

seminar and an

unpublished preprint by

^{the third author.}

Remark on the notation. - The letter C will be used to denote various

positive

^{constants either universal or}

dependent only

^on

f,

^{which may}

differ from one formula to another even within a

single

^{stream of estimates.}

D will denote the end of a

proof, .

will denote the end of a statement of a

theorem, lemma,

^etc.

1.

BASIC LEMMAS

LEMMA 1. ^- Let V be an open disc

in f

such that V n

J ~ ~

^{and V is}

disjoint

^{with a}

periodic

^orbit

Q

^of

period

^{at least 3.}

Suppose

^{we have a}

sequence of backward branches of iterates

of f-1

^on

V, namely

^holo-

morphic,

univalent functions _{gn =}

f§"’

^~n~

Iv,

^{where m}

(n)

^- 00 as n -~ 00.

Then the derivatives

g;,

converge ^to 0

uniformly

^on every compact ^{set in} V..

Proof -

^{If the lemma were} false then there would exist a

subsequence

gni ^and

points zi

^{in a} compact subset of V such

that ] &#x3E;

^{E for a}

positive

^{constant E.}

As gni

^{is a normal}

^family (the

values omit the set

Q),

there exists a

holomorphic

^{function G} which is the limit of

subsequence

of

(gn)’

^for

simplification

denote it also

by (gn)’

^{and G is} non-constant.

Then

by

the invariance and compactness of the Julia set the

point G (p) =

^{lim ’}

belongs

^to

J, provided pEV

n ^J. One of the

equivalent

definitions of _{J says} that for _every

point

^{in J and its}

neighbourhood

^{A no}

subsequence of fn IA

is normal. We conclude

by

Montel’s theorem that for

an

arbitrary

open ^set V’ such that

V’ 3 G(p)

^{and cl V’ c}

G (V),

^{for an}

arbitrary io

^{the set} U

f m (V’), covers ~

except ^{at most 2}

points.

^On

the other hand for

every i large enough

_gni

(V) ::J

^{V’ so}

f m (V’)

^C

V,

^a contradiction. D

COROLLARY 1. ^-

If U is

^a

neighbourhood ofcl(Crit+),

^{then on the}

derivatives _converge

uniformly

^{to 0}

( for

^all backward branches

LEMMA 2. - Let

p E J be a periodic point. Suppose

^{there exists a}

neigh-

bourhood V

of

p such that

(VB~ p })

^{n Crit + =}

^0.

^{Then p is a source..}

Proof -

^{If we}

supposed

^{i. e. we} would deduce that _p is a source

immediately

^from

Corollary

^1. Consider the

general

case. Let A be an open

topological

^disc

containing precisely

^one

point

qA from

O (p),

^the

periodic

^orbit

of p,

^{and such that} n Crit + ⁼

0.

Consider B a component ^{of the set}

1-1 (A) intersecting

^O

(p).

^Because

flo (p)

^is

1- to - l,

^{B contains}

precisely

^one

point

qB

belonging

^{to O}

(p).

We have

f (qB)

⁼ qA. The _map

f (BB f - ~

^{is a}

^covering

^{map to} the annulus So the fundamental _{group of}

(qA)

^{is a}

subgroup

^of

Z,

^hence

1 (qA)

^{is an}

annulus,

^{hence the} n ^{B consists} of one

point

qB

only.

^As

qB ~ Crit [otherwise

^{would be a}

sink,

^so

p ~ J], f

^{is a}

covering

map from B to the

topological

disc A. So is invertible. We have a

holomorphic

univalent branch

f v 1 :

^{A - B.} Observe that

similarly

Vol. 53, n° 4-1990.

to A also B has the property n Crit + ⁼

ø.

Indeed if

Z

Efn (Crit)

n

B.,

ⁿ &#x3E;

0, then f (Z) (Crit)

^{fl A so the}

only possibility

^is

hence a contradiction.

Now we can suppose that V is an open

topological

^disc

containing precisely

^one

point

^{from O}

(p).

^Define

by

induction branches _{gn =} on

V.

Namely having

^defined gn ^{we set} A ⁼ gn

(V)

and define gn + 1 = ¹ where

1 :

^{A ~ B is a map} considered above. Now Lemma 2 follows from the Lemma I. 0

COROLLARY 2. - For every neutral

periodic point

p

(i.

^{e. such that}

I ( f n)’ (p) I

⁼

^1,

^{where n is a}

^{period of p),}

^{there exist a critical}

^point

^{c and a}

sequence

of integers

ni ^{-~ 00} such that

(c)

^--~ p and

fni (c) ~ p..

LEMMA 3. -

Suppose

^{we have a} measurable set E c J and E &#x3E;

0,

^such

that

for

^{a. e. z ~ E there exist a} sequence

of positive integers (ni}, points

and branches

of (holomorphic, univalent) defined

^{on the}

respective

^{discs B}

(zi, E)

^which map zi ^{to z.} Then either _~,

(E)=

^{0 or there}

exists NO such that

( for

^one

of

^{the above}

sequences

(ni)}.

^.

Proof - Suppose y (E)

&#x3E; 0. Then there exists ^a

point

^of

density

^{z E}

E,

for which the

assumptions

of the lemma ^are

applicable.

^{Take a finite}

family

^of open discs

Dl,

^...,

Dm,

^{of radius}

E/4, intersecting J,

^{such that}

m

U

Dj:::::&#x3E;

^{J. For every i} there exists for which It is

j= 1

known

(topological exactness)

that there exist ^a

neighbourhood %

^{of J}

and an

integer

N &#x3E; 0 such that for

every j

^we

have f’~ (D j) :::::&#x3E;

^~.

Now

by

^Lemma

1,

definition of the

density point

and Koebe’s distortion theorem we have _Jl

(, f ’

^’~i

(Dj

^fi

^{( f}

^nt

c~.&#x3E;~

^{...-+ 1 a Hence}

again by

^{Koebe’s theorem Y}

(Dj

⁽ⁱ⁾

~ fni(E))/ (Dj(i))

^-

^1,

^hence

Jl

(4Y)

^{- 1. This}

implies

ⁱⁿ

particular

^{that J cannot be}

nowhere

dense,

^{so We obtain 1. 0}

DEFINITION. - Let us call the set

J""{

^dist

( f n (z), (ù) -+ 0 ~

^the

transverse limit set and denote it

by

^Observe

thatf-1 (~1) = c~l. 1~

LEMMA 4. -

Either ~.

^{= 0 or} ~.

(ro-1-)

=1. In the latter ^case

for

every measurable set E c with _}.l

(E)

&#x3E; 0 ^we have lim sup

(fn (E))

^{= 1..}

n -~ 00

~’roof. -

^{Let E c} Then there exist E’ ^{c E} with and ^a number 03B4&#x3E;0 such that for _every there exists ^a sequence ni ^{-+ 00} such that dist

(z), ro)

&#x3E; ð. Let N be such ^an

integer

c

B(CO,ð/2).

^So

Crit+)~2-N ö/2,

^{where 2}

is the

Lipschitz

^constant

for f

^{Fix ~ =}

-N 03B4/2

^and

apply

^{Lemma 3. 0}

2. THE CASE WHERE ALL CRITICAL POINTS IN J ARE EVENTUALLY PERIODIC

THEOREM 1. ^-

Suppose

^that every

point

^{c E Crit fl J is}

eventually periodic,

i. e. there exist n

(c) ~

^{0 such}

that fn

^(c)

(c)

^is

periodic.

^Then

(a)

every

periodic point

^{in J which is not} neutral rational

(i.

^e.

if ( f n)’ (x)

is not a root

of unity,

^{where n is a a}

period)

^{is a source.}

(b)

^the

properties (i)

^and

(ii’) from

the introduction hold..

Proof -

For every

periodic point p

ⁱⁿ

J,

^{not neutral}

rational,

^the

assumptions

^{of Lemma 2 are} satisfied. Indeed we may bother

only

^about

critical

points

^{not in L If}

eft

^{J and for a} sequence

(.n;),

^dist

(fni (c), J)

^{- 0}

then

(f"i (c))

converges ^to neutral rational

periodic point (by

the classifica- tion of components ^of

CBJ,

^the

theory

^of

^Julia,

^{Fatou and}

Sullivan).

This proves the assertion

(a).

To prove

(b)

observe that if a

periodic point p

^{E o is a source then for}

zeJ,

^{for n}

large enough.

^The

same is true if _p is neutral rational. Indeed

(P)

^{- 0 and}

f" (z) ~, f ’n (p)

for every n

imply

that for n

sufficiently large f (z) belongs

to one of the

~petals" (see [DH], Expose IX).

^{So z is in the}

domain of

normality

^{of iteration}

of f,

^{not in}

J,

^a contradiction.

(We

^owe

the last argument ^{to M.}

Lyubich.)

^{So ~1=}

J""-A

^{where A} = U

(ill)

^is

countable,

^{hence of measure} 0. The theorem follows now from Lemma 4. D

PROPOSITION 1. -

If all critical points in

^are

eventually periodic,

^then

there are no neutral

periodic points. If

all critical

points

^are

eventually periodic

^{but not}

periodic,

^{then J =}

(C

and all

periodic points

^{are sources..}

Proof -

^{If all critical}

points

^are

eventually periodic

^then

by Corollary

²

there are no neutral

points

in J. The center of a

Siegel

^{disc S is also}

excluded because as C o and oS is

uncountable,

whereas co is finite. If

additionally

^{no critical}

point

^is

periodic, if p

^{were a sink we would}

construct a normal

family

of branches

gn = fn

^{with p}

^fixed,

^{as in the}

proof

^{of the} Lemma 2. This

contradicts ~(~))~!/"(~)! 1- I -+ 00.

follows from the Fatou-Julia

theory (lack

of critical

points

^{to serve}

periodic

components ^of

CBJ)

^and Sullivan’s: no

wandering

^{domains. 0}

3. THE CASE WHERE CRITICAL POINTS ARE NOT RECURRENT Let us start with a

general

^lemma.

LEMMA 5. - For every real and an

integdr

there exists ~ &#x3E; 0 such that

for

every ⁿ and

trajectory

Vol. 53., n° 4-1990.

Proof. -

^{We can} suppose that X is

arbitrarity large

^and

^{k =1,}

^{if not,}

take an iterate instead

off

and we can compose the branches. 0

THEOREM 2. -

Suppose

^{that on the set where is}

the set

of

^all

periodic

neutral rational

points, f

^is

expanding,

^{i. e. there exist}

~, &#x3E; 1 and k such that on L we have

I (, f ’k)’ ~ &#x3E;

~,. Then the

properties (i)

^and

(ii’) from

introduction are

satisfied..

Proof. -

^{As in section 2 due to} Lemma 4 it is sufficient ^to _prove that J.1 ⁼ O. But is the union of the set U

f-n (N ~)

^countable

n&#x3E;0 so of measure

0,

and of the set

Write

apply

^{the Lemma 3} and obtain _p

(L03BB)

^{= O.}

So (L’)

^{= O. D}

Remark 1. ^- In view of Lemma

2,

^our Theorem 2 is more

general

^than

Theorem 1. One would like to prove the

expanding property

^on instead of

assuming ^it,

^{but we are not able to} do it. This is true for maps of the interval with

negative

Schwarzian derivative

provided

and there are no sinks and neutral

points,

^see

[M].

^The

reason is that for every E &#x3E; 0 there exist

k&#x3E;0,

such that if

dist ({x,

^{... ,}

^J’ M }, Crit) &#x3E;_

^s

then ^(/’ I &#x3E;

^1.

Here however the latter assertion is false even in absence of neutral

points

^{in J. A}

counterexample

^is

provided by

Michel Herman’s

example

of a

Siegel

^{disc S not}

containing

^critical

points

^{in the}

^boundary,

^see

^[D].

In this

example

^{is not}

expanding

^because

where _p is the neutral

point

in S and Q is the harmonic measure on as viewed from _p. From this

example by

Shishikura’s _surgery a counterexam-

ple

^{with Herman}

ring

^can be derived.

Another

examples

^are

provided by

^the maps of the form

/(z)==~z201420142014 2014201420142014,

^{, with r real close to 0 and see}

1- rz z - yr

[Her].

^{Such a} map is of

degree 3,

₁₌₁ ^{is a}

diffeomorphism

^{and for a}

suitable

Let us call a closed invariant set in J

disjoint

^with

Crit,

^on which

j (fk)’

&#x3E; 1 fails for

every k,

^{a neutral set. The}

question

arises what neutral sets are

possible?

Maybe

^Crit r) J ⁼

0 implies that/~~% ~ expanding?

^This

question

is related to the

question

^{whether a}

periodic

^neutral irrational

point

^can

attract a critical

point

^whose

trajectory

^never hits it and to the similar

question

^{about the}

boundary

^{of a}

Siegel disc,

^{told us}

by

^{M. Herman.}

If in the case Crit n 0) U J =

0

^a nonempty ^{neutral set different} from N f!1I exists the

question

is whether

always u({~-eJ:/"(~)~L}=0

or not?

Remark

finally

^{that for} every A ^c J such that B

(Crit, E) = 0

for _every

n = 0, 1,

^{..., if for} every x E A

then

~, (A) = 0,

^see

[L]. (The proof

is similar to that of Lemma 5 and Theorem

2, only

backward iteration should be

replaced by

the forward

one.)

So Theorem 2 holds if the

"expanding" assumption

^is

replaced by

the

formally

^{weaker one:} such that

x(/~))&#x3E;0.

^{We do not}

know however _{of any}

example exhibiting

the difference between these

assumptions..

4. ABSOLUTELY CONTINUOUS INVARIANT MEASURE

THEOREM 3. - Under the

assumptions of

^{Theorem 1 or}

2, i. f ’

~,

(J) ~ 0 (i.

^{e. J =}

C),

^{there exists an invariant measure} 11

equivalent

^to Jl ^on

t..

Proof (relying

^on the ideas from

[K-S], [M]

^and

[S]).

Vol. 53, n° 4-1990.

1"’

From the _sequence of measures y

= - ~ /~ ^(~~

^{we choose a}

^{weakly -}

^*

~ j=c

convergent

subsequence _v~i

and consider the limit measure ~ .

For every every small

positive by

Koebe’s distortion theorem

applied

^{to the}

branches f-nv

^{on the disc}

for

every A

^{c B}

(z, 8/2)

^{we have}

where C is a constant

depending only on f.

This follows

precisely

^from

cf (*)

from the introduction.

For A = B

(z, t/2)

^the

inequality ( 1 ) (the

^{left hand}

side)

^{~summed up over}

all branches

gives

the estimate

independent

^{of n.}

Now for _{.every n} we sum up the

right

^{hand side}

inequality of (1)

^over

all branches and deduce from the

resulting

^{estimate for}

f£

from

(2)

and from the definition of r; that _r; is

absolutely

continuous with respect to

(write

^on

Moreover

We want to know

that ~ ~

^on

cl (Crit + )

^{as well. We have}

(cf

^the

proof

^{of Theorem}

2).

^{So we need to} prove 11

(cl (Crit + )) = 0

^or

equivalenty

11

«(0)=0.

This will be done in Lemma 6.

Meanwhile observe

implies

^{also the}

equivalence

^of

11 and _u.

Indeed,

^{take a disc}

Bo = B (zo, Eo)

^{such that 11}

(Bo) &#x3E; 0

^and

Then

by ( 1 )

the function

I ^{log d~/d} I

^{is bounded}

on

Bo,

^so 11 and _p are

equivalent

^on

Bo.

By

^{and the}

property 11 (.~’(A) &#x3E; ~ (A)

^{for every}

^A,

^the

n z o

equivalence of 11

^{and Jl holds for every z e}

Bc1 {Crit + )

^{on a disc B}

^(z,

^E

^(z))

hence on all of

C.

⁰

LEMMA 6. -

~((D)=0. N

Proof - By

^{Lemma 5 and the}

expanding property of f on 03C9

^the

following

^{is true:}

(4)

there exist

Co, ^8,

^and À I such that for every n &#x3E; 0 and

y={~...,/"(Jc)}c:B(o,s)

there exists ^a univalent branch ^on B

~.~’’~ (x), õ),

^{to x,}

having

distortion on B

(, f’~ (x), õ)

^{less than}

2 and such that

As critical

points

^{are attracted} to co, there exists N&#x3E;0 such that U

/" (Crit)

^{c B}

(m, s/3).

^This

implies

^{that for} every ^xe

ê

if dist

(x, eo)

~

E/2,

then

dist (x,

U

/"(Crit))~~/6.

^Then

n~N

(2

^{is the}

Lipschitz

^constant

for f).

Take

b2

^{8 such that for} every disc D

in f

of

radius 2 Co 8~:

(6)

every component

of j- (D)

has diameter less than

8/2

and

(7)

every component

of (D)

^has diameter less

than 51 /2.

(Clearly

^{it suffices to take}

ð2 ~

^Const

õ1N,

^{where d is}

degree of f. )

^Now

take an

arbitrary

^{disc with}

Since (03C9) = 0

there exists a set A ^c B

(z, ~~/2) containing

^B

(z, b2/3j

^{n ill with Jl}

(A) arbitrarily

^{small and}

11

(Fr A)

^{= 0}

hence 11 (A)

^{= lim}

_v,~~ (A).

Let us Consider the set g- of all components ^of Divide ff into n families of sets as follows: for _every

k = 0, 1 ,

... , ~ let is the maximal

integer

such that for

every

nf- ~’~({z})

^U

B(~s)~0}.

Consider first k = n. Then for _every

We used the fact

(4), namely

^that

~~ (T)

^{is 1-} to - I and the

respective

branch

~’v ~‘ :

^B

(z, ð2)

^{- T} has the distortion bounded

by

^2.

Fix now k n and divide

ff k

^{into two families}

For every every reg we have

again

^the

inequality (8), only

4 may

change

^{to another constant.}

Consider now

sing’ Write

M = min (N, n - k).

Observe that

where D is

reciprocal

^{to the maximal}

degree

^{at critical}

points.

We leave the first

inequality

^{as an exercise to the}

reader;

^{the constant}

C

depends

^{on M}

(which

^{is bounded}

by N),

^{so it}

depends only

^on

f

^The

second

inequality similarly

^to

(8)

follows from the bounded distortion property

[see (4)].

If M = n - k we obtain from

(9)

Vol. 53, n° 4-1990.

Consider now k such that M = N n - k. Fix an

arbitrary

sing’

Write and A’ = B’

nf-k-N(A). ^By

^{is contained}

in a disc of radius 2

Co ð2’

^so

by (7)

^{B’ is contained in a disc B}

(y, ðl/2)

for a

point y E B’. By (6)

^and

by

the definition of

ffk

^{we So}

by (5)

^We

apply

the bounded distortion property

[cf. ( 1 )]

and obtain with the use

of

(9)

Now for _{every T}

multiply

^the both sides of the

respective inequality (8), (10)

^or

( 11 ) by ~(T),

^{then sum up these}

inequalities

^{over all T. We}

obtain

(~ (B (z, b2/2))

is swallowed

by C).

Hence ~(03C9 ~ B(z, 03B42/3))=0.

^{As z ~ 03C9 has} been chosen

arbitrarily,

^we

obtain 11

^{=0. D}

Remark 3. - Observe that

where

d = deg f,

^K denotes the number of critical

points

^{in and}

v j -

¹

are their

multiplicities.

Indeed

if 03B42

^is chosen small

enough,

then every critical

point

is contained in at most one set from the

fn - k -1 (T)

^{for every}

T E sing’ .

Remark 4. - From

(12)

^one

immediately

obtains for _{every A}

c ~

Also a related estimate, better than

(3),

^holds

Proof of (15). -

^{Let for}

z E cl (Crit+),

^{see the notation}

from the

proof

^of Lemma 6. We assumed there but the discussion there holds for _every z. If

z ~ B (m, E),

^{then for every n we have}

only decomposition

^{J =}

Jreg ~ Jsing,

^where

T ~ Jreg

^or

Jsing ^depending

^as

Denote dist and suppose

@&#x3E;0.

^{We want to estimate}

tortion property ^we have for the branch B -~ T

Suppose

^that sing and fix We have

By

the bounded distortion property ^{I x}

III ~

^{C. To estimate II write}

Crit + ) and

^denote

_by

^{D the}

reciprocal

to the

degree of fM

We have

Summing

up the

respective inequalities

^{over all} T proves

( 15).

^D

Remark 5. - One could

slightly modify

^the

proofs

^{of Lemma 6 and}

inequality ( 15)

^{if one observed that}

where

d = deg f (observe

^{that the}

right

^{hand side constant is}

independent

of k and

n).

This estimate follows from the fact that for

sing’ for s &#x3E;_ n -

k,

does not contain _any critical

point

^{and for} every

~:M2014A;2014M~~2014A;

^{at most the number}

2 d - 2 &#x3E;_ Card (Crit)

^{of com-} ponents

of f S sing)

contain critical

points.

Thus,

^{in the}

proof

^{of Lemma 6 for}

example,

^if instead of

(9)

^{we used}

the weaker

inequality

we would also succeed:

Remark 6. - There

exists ~

&#x3E; 0 such that for _every E &#x3E;

0,

This fact is stronger than Lemma 6. It follows from

( 14)

^{and the similar}

fact with 11

replaced by

J.1. The latter is a

general phenomenon. Namely

the

following

^holds:

PROPOSITION 2. - Let X be a closed nowhere dense subset

of

^such

that

f(X)

^{= X and}

f|x

^is

expanding.

^Then

for

upper limit

capacity defined

Vol. 53, n° 4-1990.

of

^{radius s}

covering X,

^{we have}

and there exist

C,

K .&#x3E; ~ such that

for

ever~y ~ &#x3E;

0,

Proof (A hint). -

^Consider

covering by

squares rather than disc and observe with the use of Koebe’s distortion theorem that there exists an

integer

^{M such that for} every square K

partitioned

^{into a}

family

^{Jf of}

squares of side where r denotes the side of

K,

^{at least one element} of Jf is

disjoint

^{with X. Next use} this observation to

subsequent

^divisions

into squares of

sides 1/Mn.

⁰

Remark 8. - Metric exactness

obviously implies ergodicity of ~.

^So

every weak - * limit of

(vn)

^is

equivalent

^to }i and

.ergodic.

^{We conclude}

that the whole sequence Vn is

weakly - *

convergent ^{to the}

unique

measure 11.

Moreover,

the sequence of the densities

F n =

^is

equiconti-

nuous on every compact ^{set K}

disjoint

^{with Crit +.} This follows from the theorem of Koebe

[H],

Lemma 17.4. I that there exists a universal con- stant C&#x3E;O such that for the disc

B(z,s) disjoint

^{with Crit +}

(cf

^the

begin

of Proof of Th.

3),

^for every

branchy"

^{we have}

Indeed this allows to estimate the derivative

along

^an

arbitrary

^{vector v}

of

length

¹

We conclude that the densities converge ^to

uniformly

^on

every K..

In fact the above convergence holds even in

analytic functions, namely

the

following

^holds:

PROPOSITION 3. - For _every disc B

(z, E) disjoint

^{with Crit + the}

functions Fn

⁼

dfn*( )/d

^{are real}

analytic

^and subharmonic.

They

^can be extended to

complex analytic functions F,~

^{on the ball}

^B (z, E/4) c ~ 2,

^{the extension}

of

the disc B

(z, E/4)

^{c f~2 =}

C , uniformly

^bounded.

In consequence ^is

real-analytic

^and subharmonic..

Proof. -

For every ^we have

where

We work in some charts in which we denote

x = (xl, x2~,

For each

branch f-nv

^we

estimate, using

^the

above,

where

So we have convergence in the

polydisc

^{in ~2} of radius

s/3. Summing of (16)

^{over all} branches

gives

^{a bound}

independent

^{of n on}

^B _{(z, ~/4~).}

So the

family Fn

^is

equicontinuous

^on compact ^{sets in}

B (z, £/4)

^and

the limit functions in the uniform convergence

topology

^are

analytic.

Subharmonicity

follows from the fact that _every

( ~ f ~ ’~)‘ ( 2~,

^{as the}

composi-

tion of the harmonic function

21og~(/~/ I

^{with the convex}

increasing

function _exp, is

subharmonic,

^see

Note :

Already

after the

completion

of this paper ^our attention was

drawn to the fact that some of our assertions are

explicitly

^{contained in}

Lyubich’s

_papers other than

[L], namely

ⁱⁿ

[11 Entropy properties

^{of rational}

endomorphisms

^{of the Riemann}

sphere, Erg.

^and

Dyn. Sys.,

^Vol.

3, 1983, pp. 351-385.

[2]

^A

study

^{of the}

stability

^of

dynamics

^{of rational}

functions,

^Teoria

Funkcij, Funkcj.

^An.

pril. 42, Kharkov, 1984, pp. 72-90 (in Russian).

[3] Dynamics

^{of rational} _{maps: the}

topological picture, Usp.

^Mat.

Nauk. ,

Vol.

41, 4, 1986, pp. 35-95.

Lemma I

overlaps

^with

[1] Prop. 3, [2]

^{Lemma 1. 1 and}

[3] Prop.

^1.10.

Corollary

^{2 can be found} in

[2]

^and

overlaps

^with

[3] Prop.

^{1.11. The} concept ^{of the}

"easy"

^set present

already

ⁱⁿ

[L]

^is

explicit

ⁱⁿ

[3]

^section

Vol. 53, n° 4-1990.

1.19. The part p

(ro-1)

^{= 0 of our} Lemma 4 and Theorem 1

[except (ii’)]

can be found there.

REFERENCES

[D] ^A. DOUADY, Disques ^de Siegel ^{et anneaux de} Herman, Séminaire Bourbaki, ^No. 677, février 1987.

[D-H] ^{A. DOUADY and J.} HUBBARD, Étude dynamique ^des polynômes complexes, ^Publ.

Math. Orsay, (2), ^Vol. 84, (4), p. 85.

[H] ^E. HILLE, Analytic ^Function Theory, Boston-New York-Chicago-Atlanta-Dallas-Palo

Alto-Toronto: Ginn and Company, ^1962.

[Her] ^M. HERMAN, Exemples de fractions rationnelles ayant ^une orbite dense sur la sphère

de Riemann, Bull. Soc. Math. France, Vol. 112, 1984, pp. 93-142.

[Hö] ^L. HÖRMANDER, ^An Introduction to Complex Analysis ^{in Several} Variables, D. van

Nostrand Company ^Inc., Princeton, New Jersey, ^1966.

[K-S] ^K. KRZYZEWSKI and W. SZLENK, On Invariant Measures for Expanding Differentiable

Mappings, ^Studia Math., Vol. 33, 1969, pp. 83-92.

[L] ^{M. Ju.} LYUBICH, On a Typical Behaviour of Trajectories ^{for a Rational} Map ^of Sphere, ^{Dokl. Ak. N. U.S.S.R., Vol.} 268.1, 1982, pp. 29-32.

[M] ^M. MISIUREWICZ, Absolutely Continuous Measures for Certain Maps ^{of an} Interval, Publications Mathématiques ^{I.H.E.S., Vol. 53,} pp. 17-51.

[R] ^M. REES, Positive Measure Sets of Ergodic ^Rational Maps, ^{Ann. Sci. Éc. Norm.}

Sup., 4e série, ^Vol. 19, 1986, pp. 383-407.

[S] ^W. SZLENK, Absolutely Continuous Invariant Measures for Rational Mappings ^of

the Sphere S2, ^Geometric Dynamics (Rio ^de Janeiro, 1981), pp. 767-783, ^Lect.

Notes Math., ^Vol. 1007, Springer, Berlin-New York, 1983.

( Manuscript ^received May 24, 1990.)